Re: proving a function is continuous

On 29 Sep 2005 08:50:48 -0700, "Artur" <artur_steiner@xxxxxxxxx>
wrote:

>Hello
>
>I'd like some hints tro prove that, if I is an [open] interval of R and f:I ->
>R is bounded on I and satisfies f((x+y)/2) <= ((f(x) + f(y))/2 for
>every x and y in I, then f is continuous.

It's easier to prove a stronger statement: If x is in I then

(*) lim sup_{y -> x} |f(x) - f(y)|/|x-y| < infinity.

Suppose that (*) fails. Then there are y_n -> x where such that
|f(x) - f(y_n)|/|x-y_n| tends to infinity. Assume that
in fact (f(x) - f(y_n))/(x-y_n) tends to infinity (you get
this or a limit of -infinity by taking a subsequence).

Note that f(x+2h) - f(x+h) >= f(x+h) - f(x) for h > 0 and
you can now show that f is unbounded.

>Thank you
>
>Artur


************************

David C. Ullrich
.

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